We have defined satisfaction in a model with a variable assignment. We have expressed formula φ{\displaystyle \varphi \,\!} being satisfied by model M{\displaystyle {\mathfrak {M}}\,\!} with variable assignment s{\displaystyle \mathrm {s} \,\!} as:

Now we can also say that a formula φ{\displaystyle \varphi \,\!} is satisfied my model M{\displaystyle {\mathfrak {M}}\,\!} (not limited to a specific variable assignment) if and only if φ{\displaystyle \varphi \,\!} is satisfied by M{\displaystyle {\mathfrak {M}}\,\!} with every variable assignment. Thus

If no free variables occur in φ,{\displaystyle \varphi ,\,\!} (thatis, if φ{\displaystyle \varphi \,\!} is a sentence), then φ{\displaystyle \varphi \,\!} is true in model M{\displaystyle {\mathfrak {M}}\,\!}.

Variable assignments allow us to deal with free variables when doing the semantic analysis of a formula. For two variable assignments, s1{\displaystyle \mathrm {s_{1}} \,\!} and s2{\displaystyle \mathrm {s_{2}} \,\!}, satisfaction by <M,s1>{\displaystyle <\!{\mathfrak {M}},\ \mathrm {s_{1}} \!>\,\!} differs from satisfaction by <M,s2>{\displaystyle <\!{\mathfrak {M}},\ \mathrm {s_{2}} \!>\,\!} only if the formula has free variables. But sentences do not have free variables. Thus a model satisfies a sentence with at least one variable assignment if and only if it satisfies the sentence with every variable assignment. The following two definitions are equivalent:

φ{\displaystyle \varphi \,\!} is true in M{\displaystyle {\mathfrak {M}}\,\!} if and only if there is a variable assignment s{\displaystyle \mathrm {s} \,\!} such that

We also noted above that for sentences (though not for formulae in general), a model satisfies the sentence with at least one variable assignment if and only if it satisfies the sentence with every variable assignment. Thus the results just listed hold for every variable assignment, not just s{\displaystyle \mathrm {s} \,\!}.

This does not require our definition of truth or the definition of satisfaction; it is simply requires evaluating the exended variable assignment. We have for any s{\displaystyle \mathrm {s} \,\!} on defined on IM{\displaystyle I_{\mathfrak {M}}\,\!}:

But F02{\displaystyle \mathrm {F_{0}^{2}} \,\!} was assigned the less then relation. Thus the preceeding holds if and only if, for every member of the domain, there is a larger member of the domain. Given that the domain is {0,1,2,...},{\displaystyle \{0,1,2,...\},\,\!} this is obviously true. Thus, (13) is true. Given that the formula of (11) and (12) is a sentence, we find the goal expressed as (11) to be met.

This holds if and only if, for every member of the domain, there is a smaller member of the domain. But there is no member of the domain smaller than 0. Thus (14) is false. The formula of (12) and (14) fails to be satisfied by M2{\displaystyle {\mathfrak {M_{2}}}\,\!} with variable assignment s{\displaystyle \mathrm {s} \,\!}. The formula of (12) and (14) is a sentence, so it fails to be satisfied by M2{\displaystyle {\mathfrak {M_{2}}}\,\!} with any variable assignment. The formula (a sentence) of (12) and (14) is false, and so the goal of (12) is met.